Existence Results for Stochastic Semilinear Differential...

Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2011, Article ID 784638, 17 pagesdoi:10.1155/2011/784638

Research ArticleExistence Results for Stochastic SemilinearDifferential Inclusions with Nonlocal Conditions

A. Vinodkumar1 and A. Boucherif2

1 Department of Mathematics and Computer Applications, PSG College of Technology, Coimbatore,Tamil Nadu 641 004, India

2 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals,P.O. Box 5046, Dhabran 31261, Saudi Arabia

Correspondence should be addressed to A. Boucherif, [email protected]

Received 31 May 2011; Accepted 6 October 2011

Academic Editor: Jiongmin Yong

Copyright q 2011 A. Vinodkumar and A. Boucherif. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

We discuss existence results of mild solutions for stochastic differential inclusions subject tononlocal conditions.We provide sufficient conditions in order to obtain a priori bounds on possiblesolutions of a one-parameter family of problems related to the original one. We, then, rely on fixedpoint theorems for multivalued operators to prove our main results.

1. Introduction

We investigate nonlocal stochastic differential inclusions (SDIns) of the form

dx(t) ∈[Ax(t) + f(t, xt)

]dt +G(t, xt)dw(t), t ∈ J = [0, T],

x(0) =m∑

i=1

γix(ti),

x(t) = ϕ(t), t ∈ J1 = (−∞, 0],

(1.1)

where T > 0, 0 < t1 < t2 < · · · < tm < T , γi are real numbers, f is a single-valued function, andG is multivalued map.

The importance of nonlocal conditions and their applications in different field havebeen discussed in [1–3]. Existence results for semilinear evolution equations with nonlocalconditions were investigated in [4–7], and the case of semilinear evolution inclusions withnonlocal conditions and a nonconvex right-hand side was discussed in [8].

2 International Journal of Stochastic Analysis

Stochastic differential equations (SDEs) play a very important role in formulation andanalysis in mechanical, electrical, control engineering and physical sciences, and economicand social sciences. See for instance [9–12] and the references therein. So far, very few articleshave been devoted to the study of stochastic differential inclusions with nonlocal conditions,see [13–15] and the references therein. Our objective is to contribute to the study of SDInswith nonlocal conditions. Motivated by the above-mentioned works and using the techniquedeveloped in [11, 16, 17], we study the SDIns of the form (1.1). The paper is organizedas follows: some preliminaries are presented in Section 2. In Section 3, we investigate theexistence of mild solutions for SDIns by using fixed point theorems for Kakutani maps.Finally in Section 4, we give an application to our abstract result.

2. Preliminaries

LetX, Y be real separable Hilbert spaces and L(Y,X) be the space of bounded linear operatorsmapping Y intoX. For convenience, we will use 〈·, ·〉 to denote inner product ofX and Y and‖ · ‖ to denote norms in X, Y , and L(Y,X) without any confusion.

Let (Ω,F, P ;F)(F = {Ft}t≥0) be a complete filtered probability space such that F0

contains all P -null sets of F. An X-valued random variable is an F-measurable functionx(t) : Ω → X and the collection of random variables H = {x(t, ω) : Ω → X : t ∈ J} iscalled a stochastic process. Generally, we just write x(t) instead of x(t, ω) and x(t) : J → X isthe space of H. Let {ei}i≥1 be a complete orthonormal basis of Y . Suppose that {w(t) : t ≥ 0}is a cylindrical Y -valued Wiener process with finite trace nuclear covariance operator Q ≥ 0,denote Tr(Q) =

∑∞i=1λi = λ < ∞, which satisfies Qei = λiei. Actually, w(t) =

∑∞i=1

√λiwi(t)ei,

where {wi(t)}∞i=1 are mutually independent one-dimensional standard Wiener processes. Weassume that Ft = σ{w(s) : 0 ≤ s ≤ t} is the σ-algebra generated by w and Ft = F. Letμ ∈ L(Y,X) and define

∥∥μ∥∥2Q = Tr

(μQμ∗) =

∞∑

n=1

∥∥∥√λnμen

∥∥∥2. (2.1)

If ‖μ‖Q < ∞, then μ is called a Q-Hilbert-Schmidt operator. Let LQ(Y,X) denote thespace of all Q-Hilbert-Schmidt operators μ : Y → X. The completion LQ(Y,X) of L(Y,X)with respect to the topology induced by the norm ‖ · ‖Q, where ‖μ‖2Q = 〈μ, μ〉 is a Hilbertspace with the above norm topology.

We now make the system (1.1) precise. Let A : X → X be the infinitesimal generatorof a compact analytic semigroup {S(t), t ≥ 0} defined on X. Let Dτ = D((−∞, 0], X) denotethe family of all right continuous functions with left-hand limit ϕ from (−∞, 0] toX and P(E)is the family of all nonempty measurable subsets of E. The functions f : [0, T] ×Dτ → X; G :[0, T]×Dτ → P(LQ(Y,X)) are Borel measurable. The phase spaceD((−∞, 0], X) is equippedwith the norm ‖φ‖ = sup−∞<θ≤0‖φ(θ)‖. We denote by Db

F0((−∞, 0], X) the family of all almost

surely bounded, F0-measurable, Dτ -valued random variables. Further, let BT be the Banachspace of allFt-adapted process φ(t,w)which is almost surely continuous in t for fixedw ∈ Ω,with norm

∥∥φ∥∥BT

=

(

sup0≤t≤T

E∥∥φ∥∥2)1/2

, (2.2)

International Journal of Stochastic Analysis 3

for any φ ∈ BT. Here the expectation E is defined by

Eχ =∫

Ωχ(w)dP. (2.3)

We shall assume throughout the remainder of the paper that the initial function ϕ ∈Db

F0((−∞, 0], X).

Some notions from set-valued analysis are in order. Denote by Pcl(X) = {Y ∈ P(X) :Y closed}, Pbd(X) = {Y ∈ P(X) : Y bounded}, Pcv(X) = {Y ∈ P(X) : Y convex}, Pcp(X) ={Y ∈ P(X) : Y compact}, Pcp,cv(X) = {Y ∈ P(X) : Y compact and convex}. A multivaluedmap F : X → P(X) is convex valued if F(x) ∈ Pcv(X) for all x ∈ X, closed valued ifF(x) ∈ Pcl(X) for all x ∈ X, F is compact valued if F(x) ∈ Pcp(X) for all x ∈ X. F is boundedon bounded sets if F(V ) = ∪x∈V F(x) is bounded in X, for all V ∈ Pbd(X); that is,

supx∈V

{sup

{∥∥y∥∥ : y ∈ F(x)

}}<∞. (2.4)

F is called upper semicontinuous (u.s.c) on X, if for each x0 ∈ X, the set F(x0) is non-empty, closed subset of X, and if for each open set V of X containing F(x0) there exists anopen neighborhoodN of x0 such that F(N) ⊆ V .

F is said to be completely continuous if F(V ) is relatively compact, for every V ∈Pbd(X).

If the multivalued map F is completely continuous with nonempty compact values,then F is u.s.c if and only if F has a closed graph (ie., xn → x∗, yn → y∗, yn ∈F(xn) imply y∗ ∈ F(x∗)).

F has a fixed point if there is x ∈ X such that x ∈ F(x). The fixed point set of themultivalued operator F will be denoted by Fix F.

The Hausdorff metric on Pbd,cl(X) is the function H : Pbd,cl(X) × Pbd,cl(X) → R+

defined by

H(A,B) = max

{

supa∈A

d(a,B), supa∈B

d(A, b)

}

, (2.5)

where d(A, b) = inf{‖a − b‖2, a ∈ A}, d(a,B) = inf{‖a − b‖2, b ∈ B}.The multivalued mapM : [0, T]Pbd,cl(X) is said to be measurable if for each x ∈ X the

function ζ : [0, T] → R+ defined by

ζ(t) = d(x,M(t)) = inf{‖x − z‖2 : z ∈M(t)

}is measurable. (2.6)

For more details on multivalued maps see [18–20]. Our existence results are based onthe following fixed point theorem (nonlinear alternative) for Kakutani maps [21].

Theorem 2.1. Let X be a Hilbert space, C a closed convex subset of X, Y an open subset of C and0 ∈ Y . Suppose that F : Y → Pcl,cv(C) is an upper semicontinuous compact map. Then either (i) Fhas a fixed point in Y or (ii) there are v ∈ ∂Y and λ ∈ (0, 1) with v ∈ λF(v).


Definition 2.2. The multivalued map G : J ×Dτ → P(LQ(Y,X)) is said to be L2-Caratheodoryif

(i) t �→ G(t, u) is measurable for each u ∈ Dτ ;

(ii) u �→ G(t, u) is upper semicontinuous for almost all t ∈ J ;(iii) for each q > 0, there exists ωq ∈ L2(J,R+) such that

‖G(t, u)‖2 := sup{∥∥g

∥∥2 : g ∈ G(t, u)}≤ ωq(t), (2.7)

for all ‖u‖2BT≤ q and for a.e. t ∈ J .

For each x ∈ L2(LQ(Y,X)) define the set of selections of G by

SG,x ={g ∈ L2 = L2(LQ(Y,X)

): g(t) ∈ G(t, xt) for a.e, t ∈ J

}. (2.8)

Lemma 2.3 (see [22]). Let I be a compact interval and X be a Hilbert space. Let G be anL2-Caratheodory multivalued map with SG,x /=φ and let Γ be a linear continuous mapping fromL2(I, X) → C(I, X). Then the operator

Γ ◦ SG : C(I, X) �−→ Pbd,cl,cv(C(I, X)), x �−→ (Γ ◦ SG)(x) = Γ(SG,x), (2.9)

is a closed graph operator in C(I, X) × C(I, X).

Definition 2.4. A semigroup {S(t), t ≥ 0} is said to be uniformly bounded if there exists aconstantM ≥ 1 such that

‖S(t)‖ ≤M, for t ≥ 0. (2.10)

Assume that

m∑

i=1

∣∣γi∣∣ <

1M

. (2.11)

Then there exists a bounded operator B on D(B) = X given by the formula

B =

(

I −m∑

i=1

γiT(ti)

)−1

. (2.12)

Definition 2.5. A stochastic process {x(t) ∈ BT, t ∈ (−∞, T]} is called a mild solution of system(1.1) if

(i) x(t) is Ft-adapted with∫T0 ‖x(t)‖2 dt <∞ almost surely;


(ii) x(t) satisfies the integral equation

x(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

ϕ(t), t ∈ J1,m∑

i=1

γiS(t)B

[∫ ti

0S(ti − s)f(s, xs)ds +

∫ ti

0S(ti − s)g(s)dw(s)

]

+∫ t

0S(t − s)f(s, xs)ds +

∫ t

0S(t − s)g(s)dw(s), a.e. t ∈ J,

(2.13)

where g ∈ SG,x.

3. Existence Results

In this section, we discuss the existence of mild solutions of the system (1.1). We need thefollowing hypotheses.

(H1): The function f : J × Dτ → X is continuous and there exist two positive constantsC1, C2 such that

∥∥f(t, xt)∥∥2 ≤ C1‖x‖2 + C2, for each x ∈ Dτ, t ∈ J. (3.1)

(H2) : G : J × Dτ → P(LQ(Y,X)) is an L2-Caratheodory multivalued function withcompact and convex values.

(H3): There exists a continuous nondecreasing function ψ : R+ → (0,∞) and p ∈L1(J,R+) such that

‖G(t, xt)‖2 = sup{∥∥g

∥∥2 : g ∈ G(t, xt)}≤ p(t)ψ

(‖x‖2

), a. et ∈ J, all x ∈ Dτ. (3.2)

Theorem 3.1. Assume that (H1)–(H3) hold. Then the system (1.1) has at least one mild solution on(−∞, T], provided that

3K1C1T < 1, supρ∈[0,∞)

{1 − 3TK1C1}ρ3TK2C2 + 3K1Tr(Q)

∥∥p∥∥L1ψ(ρ) > 1, (3.3)

where

K1 = 3(2mM2

∑m

i=1

∣∣γi∣∣2‖B‖2 + 1

)M2, K2 =

(2mM2

∑m

i=1

∣∣γi∣∣2‖B‖2 + T

)M2. (3.4)


Proof. Transform the system (1.1) into a fixed point problem. Consider the multivaluedoperator M : BT → P(BT) defined by

M(x) = h ∈ BT : h(t)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

ϕ(t), t ∈ J1m∑

i=1

γiS(t)B

[∫ ti


∫ ti


]

+∫ t


∫ t

0S(t − s)g(s)dw(s), g ∈ SG,x, a. e. t ∈ J.

(3.5)

It is clear that the fixed points ofM are mild solutions of system (1.1). Hence we have to findsolutions of the inclusion y ∈ M(y). We show that the multivalued operator M satisfies allthe conditions of Theorem 2.1. The proof will be given in several steps.

Step 1. M(x) is convex for each x ∈ BT. Since G has convex values it follows that SG,x isconvex; so that if g1, g2 ∈ SG,x then αg1 + (1 − α)g2 ∈ SG,x, which implies clearly that M(x) isconvex.

Step 2. The operator M is bounded on bounded subsets of BT. For q > 0 let Bq = {x ∈ BT :‖x‖BT

≤ q} be a bounded subset of BT. We show that M(Bq) is a bounded subset of BT. Foreach x ∈ Bq let h ∈ M(x). Then there exists g ∈ SG,x such that for each t ∈ J we have

h(t) =m∑

i=1

γiS(t)B

[∫ ti


∫ ti


]

+∫ t


∫ t

0S(t − s)g(s)dw(s),

(3.6)

‖h(t)‖2 ≤ 3

∥∥∥∥∥

m∑

i=1

γiS(t)B

[∫ ti


∫ ti


]∥∥∥∥∥

2

+ 3

∥∥∥∥∥

∫ t

0S(t − s)f(s, xs)ds

∥∥∥∥∥

2

+ 3

∥∥∥∥∥

∫ t

0S(t − s)g(s)dw(s)

∥∥∥∥∥

2

≤ 6mm∑

i=1

∣∣γi∣∣2M2‖B‖2M2

[∫ ti

0

∥∥f(s, xs)∥∥2ds + Tr(Q)

∫ ti

0

∥∥g(s)∥∥2ds

]

+ 3M2∫ t

0

∥∥f(s, xs)∥∥2ds + 3M2 Tr(Q)

∫ t

0

∥∥g(s)∥∥2ds

≤ 3

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + 1

)

M2∫T

0

∥∥f(s, xs)∥∥2ds


+ 3

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + 1

)

Tr(Q)M2∫T

0

∥∥g(s)∥∥2ds

≤ 3M2

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + 1

)(∫T

0

∥∥f(s, xs)∥∥2ds + Tr(Q)

∫T

0

∥∥g(s)∥∥2ds

)

≤ 3M2

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + 1

)

×(∫T

0

(C1‖x(s)‖2 + C2

)ds + Tr(Q)

∫T

0p(s)ψ

(‖x(s)‖2

)ds

)

≤ K1

(TC1q

2 + TC2 + Tr(Q)ψ(q2)∥∥p

∥∥L1

).

(3.7)

Hence for each h ∈ M(Bq), we get

‖h‖2BT= sup

t∈[0,T]E‖h‖2 ≤ K1T

(TC1q

2 + TC2 + Tr(Q)ψ(q2)∥∥p

∥∥L1

). (3.8)

Then, for each h ∈ M(x), we have ‖h‖2BT≤ ∧, where ∧ := K1T(TC1q

2+TC2+Tr(Q)ψ(q2)‖p‖L1).

Step 3. M sends bounded sets into equicontinuous sets in BT. For each x ∈ Bq let h ∈ M(x)be given by (3.6). Let τ1, τ2 ∈ J with 0 < τ1 < τ2 ≤ T . Then

h(τ2) − h(τ1) =m∑

i=1

γi[S(τ2) − S(τ1)]B[∫ ti


∫ ti


]

+∫ τ2

0S(τ2 − s)f(s, xs)ds +

∫ τ2

0S(τ2 − s)g(s)dw(s)

−∫ τ1

0S(τ1 − s)f(s, xs)ds −

∫ τ1

0S(τ1 − s)g(s)dw(s).

(3.9)

This implies that

h(τ2) − h(τ1) =m∑

i=1

γi[S(τ2) − S(τ1)]B[∫ ti


∫ ti


]

+∫ τ1

0[S(τ2 − s) − S(τ1 − s)]f(s, xs)ds +

∫ τ1

0[S(τ2 − s) − S(τ1 − s)]g(s)dw(s)

+∫ τ2

τ1

S(τ2 − s)f(s, xs)ds +∫ τ2

τ1

S(τ2 − s)g(s)dw(s).

(3.10)


It follows that

‖h(τ2) − h(τ1)‖2 ≤ 5m‖B‖2‖S(τ2) − S(τ1)‖2m∑

i=1

∣∣γi∣∣2∫ ti

0‖S(ti − s)‖2


+ 5m‖B‖2‖S(τ2) − S(τ1)‖2m∑

i=1

∣∣γi∣∣2 Tr(Q)

∫ ti

0‖S(ti − s)‖2

∥∥g(s)∥∥2ds

+ 5∫ τ1

0‖S(τ2 − s) − S(τ1 − s)‖2


+ 5∫ τ2

τ1

‖S(τ2 − s)‖2∥∥f(s, xs)

∥∥2ds

+ 5Tr(Q)∫ τ1

0‖S(τ2 − s) − S(τ1 − s)‖2

∥∥g(s)∥∥2ds

+ 5Tr(Q)∫ τ2

τ1

‖S(τ1 − s)‖2∥∥g(s)

∥∥2ds

≤ 5mM2‖B‖2‖S(τ2) − S(τ1)‖2m∑

i=1

∣∣γi∣∣2{(C1q

2 + C2

)+ Tr(Q)ψ

(q2)∥∥p

∥∥L1

}

+ 5(C1q

2 + C2

)∫ τ1

0‖S(τ2 − s) − S(τ1 − s)‖2ds

+ 5M2(C1q

2 + C2

)(τ2 − τ1)

+ 5Tr(Q)ψ(q2)∫ τ1

0‖S(τ2 − s) − S(τ1 − s)‖2p(s)ds

+ 5M2 Tr(Q)ψ(q2)∥∥p

∥∥L1(τ2 − τ1).

(3.11)

Since there is δ > 0 such that

‖S(τ2) − S(τ1)‖ ≤ δ√τ1

√τ2 − τ1, (3.12)

(see [23, proposition 1]) and the compactness of S(t) for t > 0 implies the continuity in theuniform operator topology, we have

‖S(τ2) − S(τ1)‖2 −→ 0, ‖S(τ2 − s) − S(τ1 − s)‖2 −→ 0 as τ2 −→ τ1. (3.13)

Therefore

E‖h(τ2) − h(τ1)‖2 −→ 0 as τ2 −→ τ1. (3.14)


When τ1 = 0 we have

‖h(τ2) − h(0)‖2 ≤ 5mM2‖B‖2‖S(τ2) − S(0)‖2m∑

i=1

∣∣γi∣∣2{ti(C1q

2 + C2

)+ Tr(Q)ψ

(q2)∥∥p

∥∥L1

}

+ 5M2(C1q

2 + C2

)τ2 + 5M2 Tr(Q)ψ

(q2)∥∥p

∥∥L1τ2,

(3.15)

so that, similar to the previous situation, we have

E‖h(τ2) − h(0)‖2 −→ 0 as τ2 −→ 0. (3.16)

Step 4. M sends bounded sets into relatively compact sets in BT. Let 0 < ε < t, for t ∈ J . Forx ∈ Bq define a function hε by

hε(t) =m∑

i=1

γiS(t)B

[∫ ti


∫ ti


]

+∫ t−ε


∫ t−ε


(3.17)

where g ∈ SG,x. Since S(t) is a compact operator, the set Vε(t) = {hε(t) : hε ∈ M(x)} isrelatively compact in BT for every ε in (0, t). Moreover, for every h ∈ M(x)we have

E‖h − hε‖2 ≤ 2εM2∫ t

t−ε

[C1E‖x(s)‖2 + C2

]ds + 2M2 Tr(Q)

∫ t

t−εωq(s)ds

≤ 2ε2M2(C1q + C2)+ 2M2 Tr(Q)

∫ t

t−εωq(s)ds.

(3.18)

Since ωq ∈ L1(J) and meas([t − ε, t]) = ε it follows that

‖h − hε‖BT−→ 0 as ε −→ 0. (3.19)

As a consequence of Step 1 through Step 4, together with Ascoli-Arzela theorem, we canconclude that the multivalued operator M is compact.

Step 5. M has a closed graph. Let xn → x∗ and hn ∈ M(xn) with hn → h∗. We shall showthat h∗ ∈ M(x∗).

There exists gn ∈ SG,xn such that

hn(t) =m∑

i=1

γiS(t)B

[∫ ti

0S(ti − s)f(s, xn,s)ds +

∫ ti

0S(ti − s)gn(s)dw(s)

]

+∫ t

0S(t − s)f(s, xn,s)ds +

∫ t

0S(t − s)gn(s)dw(s).

(3.20)


We must prove that there exists g∗ ∈ SG,x∗ such that

h∗(t) =m∑

i=1

γiS(t)B

[∫ ti

0S(ti − s)f(s, x∗

s)ds +∫ ti

0S(ti − s)g∗(s)dw(s)

]

+∫ t

0S(t − s)f(s, x∗

s)ds +∫ t

0S(t − s)g∗(s)dw(s).

(3.21)

Consider the linear continuous operator Γ : L2(LQ(Y,X)) → BT defined by

Γ(g)(t) =

m∑

i=1

γiS(t)B∫ ti

0S(ti − s)g(s)dw(s) +

∫ t

0S(t − s)g(s)dw(s). (3.22)

Clearly, Γ is linear and continuous. Indeed, one has

∥∥Γ(g)(t)∥∥2 ≤

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + 1

)

Tr(Q)M2∫ t

0

∥∥g(s)∥∥2ds

E∥∥Γ(g)∥∥2 ≤

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + 1

)

Tr(Q)M2∥∥ωq

∥∥L1 .

(3.23)

Let

Θn(t) = hn(t) −m∑

i=1

γiS(t)B∫ ti

0S(ti − s)f(s, xn,s)ds −

∫ t

0S(t − s)f(s, xn,s)ds,

Θ∗(t) = h∗(t) −m∑

i=1

γiS(t)B∫ ti

0S(ti − s)f(s, x∗

s)ds −∫ t

0S(t − s)f(s, x∗

s)ds.

(3.24)

We have

Θn(t) ∈ Γ ◦ SG,xn . (3.25)

Since f is continuous (see (H1))

‖Θn(t) −Θ∗(t)‖2 −→ 0 as n −→ ∞. (3.26)

Lemma 2.3 implies that Γ ◦ SG has a closed graph. Hence there exists g∗ ∈ SG,x∗ such that

Θ∗(t) =m∑

i=1

γiS(t)B∫ ti

0S(ti − s)g∗(s)dw(s) +

∫ t

0S(t − s)g∗(s)dw(s). (3.27)

Hence h∗ ∈ M(x∗), which shows that graph M is closed.


Step 6. Let λ ∈ (0, 1) and let x ∈ λM(x). Then there exists g ∈ SG,x such that

x(t) = λm∑

i=1

γiS(t)B

[∫ ti


∫ ti


]

+ λ∫ t

0S(t − s)f(s, xs)ds + λ

∫ t

0S(t − s)g(s)dw(s).

(3.28)

Thus

‖x(t)‖2 ≤ 3

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + T

)

M2∫ t

0


+ 3

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + 1

)

M2 Tr(Q)∫ t

0

∥∥g(s)∥∥2ds.

(3.29)

Conditions (H1)–(H3) imply that for each t ∈ J

E‖x‖2 ≤ 3

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + T

)

M2∫ t

0

[C1E‖x(s)‖2 + C2

]ds

+ 3

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + 1

)

M2 Tr(Q)∫ t

0p(s)ψ

(E‖x(s)‖2

)ds.

(3.30)

The function � defined on [0, T] by

�(t) = sup{E‖x(s)‖2 : 0 ≤ s ≤ t

}(3.31)

satisfies

�(t) ≤ 3T

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + T

)

M2C2 + 3T

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + T

)

M2C1�(t)

+ 3

(

2mM2m∑

i=1

∣∣γi∣∣2‖B‖2 + 1

)

M2 Tr(Q)∥∥p∥∥L1ψ(�(t)

).

(3.32)

This yields

�(t) ≤3TK2C2 + 3K1 Tr(Q)

∥∥p∥∥L1ψ(�(t)

)

1 − 3TK1C1. (3.33)

Since

‖x‖BT= sup

0≤t≤T�(t), (3.34)


it follows that

‖x‖BT≤

3TK2C2 + 3K1 Tr(Q)∥∥p∥∥L1ψ(‖x‖BT

)

1 − 3TK1C1. (3.35)

Therefore

(1 − 3TK1C1)‖x‖BT

3TK2C2 + 3K1 Tr(Q)∥∥p∥∥L1ψ(‖x‖BT

) ≤ 1. (3.36)

Now, by (3.3) there exists ρ0 > 0 such that

{1 − 3TK1C1}ρ03TK2C2 + 3K1 Tr(Q)

∥∥p∥∥L1ψ(ρ0) > 1. (3.37)

LetY = {v ∈ BT : ‖v‖BT< ρ0}. Suppose that there is v ∈ ∂Y such that v ∈ λM(v) for λ ∈ (0, 1).

Then ‖x‖BT= �0 satisfies (3.36), which contradicts (3.37). So, alternative (ii) in Theorem 2.1.

does not hold, and consequently, the multivalued operator M has a fixed point, which is asolution of (1.1).

We now present another existence result for system (1.1). We shall assume that thesingle-valued f and the multivalued G satisfy a Wintner-type growth condition with respectto their second variable.

Theorem 3.2. Assume that (H2) and the following condition hold.

(HfG): There exists � ∈ L1([0, T],R+) such that

H(f(t, xt), f

(t, yt

))∨H

(G(t, xt), G

(t, yt

))≤ �(t)

∥∥x − y∥∥2, ∀t ∈ J, x, y ∈ Dτ,

H(0, f(t, 0)

)∨H(0, G(t, 0)) ≤ �(t), a.e. t ∈ J,

(3.38)

then the system (1.1) has at least one mild solution on (−∞, T].

Remark 3.3. H(f(t, xt), f(t, yt)) = ‖f(t, xt) − f(t, yt)‖2.

Proof. Themultivalued operatorM defined in the proof of the previous theorem is completelycontinuous and upper semicontinuous. Now, we prove that

Y = {x ∈ BT : x ∈ λM(x) for some λ ∈ (0, 1)} (3.39)


is bounded. Let x ∈ Y. Then there exists g ∈ SG,x such that for each t ∈ J

x(t) = λm∑

i=1

γiS(t)B

[∫ ti


∫ ti


]

+ λ∫ t

0S(t − s)f(s, xs)ds + λ

∫ t


(3.40)

for some λ ∈ (0, 1). Then

‖x(t)‖2 ≤ 3

(

2mM2m∑

i=1

ti∥∥γi∥∥2‖B‖2 + T

)

M2∫ t

0


+ 3

(

2mM2m∑

i=1

∥∥γi∥∥2‖B‖2 + 1

)

M2 Tr(Q)∫ t

0

∥∥g(s)∥∥2ds

≤ 6

(

2mM2m∑

i=1

ti∥∥γi∥∥2‖B‖2 + T

)

M2∫ t

0�(s)

(1 + ‖x(s)‖2

)ds

+ 6

(

2mM2m∑

i=1

∥∥γi∥∥2‖B‖2 + 1

)

M2 Tr(Q)∫ t

0�(s)

(1 + ‖x(s)‖2

)ds.

(3.41)

Thus

E‖x(t)‖2 ≤ Q1 +Q2

∫ t

0�(s)E‖x(s)‖2ds, (3.42)

where

Q1 = 6

(

2mM2m∑

i=1

ti∥∥γi∥∥2‖B‖2 + T

)

M2(T + Tr(Q))‖�‖L1 ,

Q2 = 6

(

2mM2m∑

i=1

∥∥γi∥∥2‖B‖2 + 1

)

M2(T + Tr(Q)).

(3.43)

Using the function �(t), defined by (3.31), we obtain

�(t) ≤ Q1 +Q2

∫ t

0�(s)�(s)ds. (3.44)

Gronwall’s inequality gives

�(t) ≤ Q1 exp(Q2‖�‖L1), ∀t ∈ J. (3.45)


Therefore there exists β > 0 such that

�(t) ≤ β, ∀t ∈ J, (3.46)

which implies that

‖x‖2BT≤ β. (3.47)

This shows that Y is bounded. Theorem 2.1. shows that M has a fixed point, which is asolution of (1.1), and this completes the proof.

4. Example

Consider the following stochastic partial differential inclusion with infinite delay

∂

∂tv(t, x) ∈

n∑

i,j=1

∂

∂xi

(

aij(x)∂

∂xjv(t, x)

)

− a0v(t, x) + εn∑

i=1

∂

∂xiv(t − r, x)

+∫0

−∞β1(θ)v(t + θ, x)dθ +

∫0

−∞β2(t, x, θ)G1(v(t + θ, x))dθdβ(t)

v(t, x) = 0, t ∈ J, x ∈ ∂Δ,

v(0, x) =n∑

i=1

βk(x)v(x, tk), x ∈ Δ, tk ∈ [0, T],

v(θ, x) = ϕ(0, x), −∞ < θ ≤ 0, x ∈ Δ,

(4.1)

where a0, r, and ε are positive constants, J = [0, T], Δ is an open bounded set in Rn

with a smooth boundary ∂Δ, β1 : (−∞, 0] → R is a positive function, β(t) stands for astandard cylindrical Wiener process in L2(Δ) defined on a stochastic basis (Ω,F, P), andϕ ∈ Db

F0((−∞, 0], L2(Δ)).The coefficients aij ∈ L∞(Δ) are symmetric and satisfy the ellipticity condition

n∑

i,j=1

aij(x)ξiξj ≥ κ|ξ|2, x ∈ Δ, ξ ∈ Rn, (4.2)

for a positive constant κ.In order to rewrite (4.1) in the abstract form, we introduce X = L2(Δ) and we define

the linear operator A : D(A) ⊂ X → X by

D(A) = H2(Δ) ∩H10(Δ); A = −

n∑

i,j

∂

∂xi

(

aij(x)∂

∂xj

)

. (4.3)


HereH1(Δ) is the Sobolev space of functions u ∈ L2(Δ)with distributional derivativeu′ ∈ L2(Δ),H1

0(Δ) = {u ∈ H1(Δ); u = 0 on ∂Δ} andH2(Δ) = {u ∈ L2(Δ); u′, u′′ ∈ L2(Δ)}.Then A generates a symmetric compact analytic semigroup e−tA in X, and there exists

a constantM1 > 0 such that ‖e−tA‖ ≤M1. Also, note that there exists a complete orthonormalset {ξn}, (n = 1, 2, . . .) of eigenvectors of Awith ξn(x) =

√(2/n) sin(nx).

We assume the following conditions hold.

(i) The function β1(·) is continuous in J with

∫0

−∞β1(θ)

2dθ <∞. (4.4)

(ii) The function β2(·) ≥ 0 is continuous in J ×Δ × (−∞, 0)with

∫0

−∞β2(t, x, θ)dθ = p1(t, x) <∞,

(∫

Δp21(t, x)dx

)1/2

<∞. (4.5)

(iii) The multifunction G1(·) is an L2-Caratheodory multivalued function with compactand convex values and

0 ≤ ‖G1(v(θ, x))‖ ≤ ψ0(‖v(θ, ·)‖L2), (θ, x) ∈ J ×Δ, (4.6)

where ψ0(·) : [0,∞) → (0,∞) is continuous and nondecreasing.

Assuming that conditions (i)–(iii) are verified, then the problem (4.1) can be modeledas the abstract stochastic partial functional differential inclusions of the form (1.1), with

f(t, vt) =∫0

−∞β1(θ)v(t + θ, x)dθ

G(t, vt) =∫0

−∞β2(t, x, θ)G1(v(t + θ, x))dθ, γi = βk(x).

(4.7)

The next result is a consequence of Theorem 3.1.

Proposition 4.1. Assume that the conditions (i)–(iii) hold. Then there exists at least one mild solutionv for the system (4.1) provided that

supρ∈[0,∞)

{1 − 3TK2C1}ρ3K1Tr(Q)

∥∥p∥∥L1ψ0

(ρ) > 1, (4.8)

where K1 = (2mM21∑m

i=1‖γi‖2‖B‖2 + 1)M2

1 and K2 = (2mM21∑m

i=1ti‖γi‖2‖B‖2 + T)M2

1.

Proof. Condition (i) implies that (H1) holds with C1 =∫0−∞ β21(θ)dθ and C2 = 0. (H2) and (H3)

follow from conditions (ii) and (iii)with p(t) = (∫Δ p

21(t, x)dx)

1/2 and ψ = ψ0.


Acknowledgments

A. Boucherif is grateful to King Fahd University of Petroleum and Minerals for its constantsupport. The authors sincerely thank the anonymous reviewer for his/her constructivecomments and suggestions to improve the quality of the paper.

References

[1] K. Balachandran and K. Uchiyama, “Existence of solutions of nonlinear integrodifferential equationsof Sobolev typewith nonlocal condition in Banach spaces,” Proceedings of the Indian Academy of Sciences(Mathematical Sciences), vol. 110, no. 2, pp. 225–232, 2000.

[2] L. Byszewski, “Theorems about the existence and uniqueness of solutions of a semilinear evolutionnonlocal Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 162, no. 2, pp. 494–505, 1991.

[3] K. Deng, “Exponential decay of solutions of semilinear parabolic equations with nonlocal initialconditions,” Journal of Mathematical Analysis and Applications, vol. 179, no. 2, pp. 630–637, 1993.

[4] M. Benchohra, E. P. Gatsori, J. Henderson, and S. K. Ntouyas, “Nondensely defined evolutionimpulsive differential inclusions with nonlocal conditions,” Journal of Mathematical Analysis andApplications, vol. 286, no. 1, pp. 307–325, 2003.

[5] A. Boucherif and R. Precup, “Semilinear evolution equations with nonlocal initial conditions,”Dynamic Systems and Applications, vol. 16, no. 3, pp. 507–516, 2007.

[6] X. Fu and K. Ezzinbi, “Existence of solutions for neutral functional differential evolution equationswith nonlocal conditions,” Nonlinear Analysis, vol. 54, no. 2, pp. 215–227, 2003.

[7] J. Liang and Ti.-J. Xiao, “Semilinear integrodifferential equations with nonlocal initial conditions,”Computers & Mathematics with Applications, vol. 47, no. 6-7, pp. 863–875, 2004.

[8] A. Boucherif, “Semilinear evolution inclusions with nonlocal conditions,” Applied Mathematics Letters,vol. 22, no. 8, pp. 1145–1149, 2009.

[9] N. U. Ahmed, “Nonlinear stochastic differential inclusions on Banach space,” Stochastic Analysis andApplications, vol. 12, no. 1, pp. 1–10, 1994.

[10] P. Balasubramaniam and S. K. Ntouyas, “Controllability for neutral stochastic functional differentialinclusions with infinite delay in abstract space,” Journal of Mathematical Analysis and Applications, vol.324, no. 1, pp. 161–176, 2006.

[11] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44, Cambridge UniversityPress, Cambridge, UK, 1992.

[12] P. Kree, “Diffusion equation for multivalued stochastic differential equations,” Journal of FunctionalAnalysis, vol. 49, no. 1, pp. 73–90, 1982.

[13] S. Aizicovici and V. Staicu, “Multivalued evolution equations with nonlocal initial conditions inBanach spaces,” Nonlinear Differential Equations and Applications, vol. 14, no. 3-4, pp. 361–376, 2007.

[14] D. N. Keck and M. A. McKibben, “Functional integro-differential stochastic evolution equations inHilbert space,” Journal of Applied Mathematics and Stochastic Analysis, vol. 16, no. 2, pp. 141–161, 2003.

[15] A. Lin and L. Hu, “Existence results for impulsive neutral stochastic functional integro-differentialinclusions with nonlocal initial conditions,” Computers & Mathematics with Applications, vol. 59, no. 1,pp. 64–73, 2010.

[16] A. Elazzouzi and A. Ouhinou, “Optimal regularity and stability analysis in the α-norm for a classof partial functional differential equations with infinite delay,” Discrete and Continuous DynamicalSystems, vol. 30, no. 1, pp. 115–135, 2011.

[17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44, Springer,New York, NY, USA, 1983.

[18] J.-P. Aubin and A. Cellina, Differential Inclusions, vol. 264, Springer, Berlin, Germany, 1984.[19] K. Deimling,Multivalued Differential Equations, vol. 1, Walter de Gruyter, New York, NY, USA, 1992.[20] S. Hu and N. S. Papageorgiou, “On the existence of periodic solutions for nonconvex-valued

differential inclusions in Rn,” Proceedings of the American Mathematical Society, vol. 123, no. 10, pp.

3043–3050, 1995.[21] A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, NY, USA, 2003.


[22] A. Lasota and Z. Opial, “An application of the Kakutani-Ky-Fan theorem in the theory of ordinarydifferential equations,” Bulletin de l’Academie Polonaise des Sciences. Serie des Sciences Mathematiques,Astronomiques et Physiques, vol. 13, pp. 781–786, 1965.

[23] J. Hofbauer and P. L. Simon, “An existence theorem for parabolic equations onRN with discontinuous

nonlinearity,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 8, pp. 1–9, 2001.

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